Below, we outline the characteristics of the Edgeworth expansion of \(T_n\) , conditioned on \(\mathcal {F}_n\) . Clearly, the classical Edgeworth expansion (see e.g. [1]}) needs to be adjusted in terms of approximating polynomials and functions. In particular, for \(l,r,m\in \mathbb {N}\)  and \(t\in \mathbb {R}\) , we define
\(\widetilde{\lambda }_{l,n}&=\Big (\sum \widetilde{\mu }_{2,j}\Big )^{-l/2}\sum \widetilde{\kappa }_{l,j}=V_n^{-l}\sum \widetilde{\kappa }_{l,j}\quad (\widetilde{\lambda }_{l,n}=0 \text{ if } V_n=0),\\\widetilde{U}_{r,n}(it)&=\sum _{*(k_\cdot ,r,\cdot )} (it)^{r+2u(k_\cdot )} \prod _{l=1}^{r} \frac{1}{k_l!} \Big (\frac{\widetilde{\lambda }_{l+2,n}}{(l+2)!}\Big )^{k_l},\\\widetilde{\varphi }_{\Phi ^{\widetilde{P}}_{m,n}}(t)&=e^{-t^2/2}\Big (1+\sum _{r=1}^{m-2} \widetilde{U}_{r,n}(it)\Big ).\) 
